A.5.3.1 Floating Point Representation Concepts

This section introduces the terminology for describing floating point
representations.

You are probably already familiar with most of these concepts in terms
of scientific or exponential notation for floating point numbers. For
example, the number 123456.0 could be expressed in exponential
notation as 1.23456e+05, a shorthand notation indicating that the
mantissa 1.23456 is multiplied by the base 10 raised to
power 5.

More formally, the internal representation of a floating point number
can be characterized in terms of the following parameters:

The sign is either -1 or 1.

The base or radix for exponentiation, an integer greater
than 1. This is a constant for a particular representation.

The exponent to which the base is raised. The upper and lower
bounds of the exponent value are constants for a particular
representation.

Sometimes, in the actual bits representing the floating point number,
the exponent is biased by adding a constant to it, to make it
always be represented as an unsigned quantity. This is only important
if you have some reason to pick apart the bit fields making up the
floating point number by hand, which is something for which the GNU C Library
provides no support. So this is ignored in the discussion that
follows.

The mantissa or significand is an unsigned integer which is a
part of each floating point number.

The precision of the mantissa. If the base of the representation
is b, then the precision is the number of base-b digits in
the mantissa. This is a constant for a particular representation.

Many floating point representations have an implicit hidden bit in
the mantissa. This is a bit which is present virtually in the mantissa,
but not stored in memory because its value is always 1 in a normalized
number. The precision figure (see above) includes any hidden bits.

Again, the GNU C Library provides no facilities for dealing with such
low-level aspects of the representation.

The mantissa of a floating point number represents an implicit fraction
whose denominator is the base raised to the power of the precision. Since
the largest representable mantissa is one less than this denominator, the
value of the fraction is always strictly less than 1. The
mathematical value of a floating point number is then the product of this
fraction, the sign, and the base raised to the exponent.

We say that the floating point number is normalized if the
fraction is at least 1/b, where b is the base. In
other words, the mantissa would be too large to fit if it were
multiplied by the base. Non-normalized numbers are sometimes called
denormal; they contain less precision than the representation
normally can hold.

If the number is not normalized, then you can subtract 1 from the
exponent while multiplying the mantissa by the base, and get another
floating point number with the same value. Normalization consists
of doing this repeatedly until the number is normalized. Two distinct
normalized floating point numbers cannot be equal in value.

(There is an exception to this rule: if the mantissa is zero, it is
considered normalized. Another exception happens on certain machines
where the exponent is as small as the representation can hold. Then
it is impossible to subtract 1 from the exponent, so a number
may be normalized even if its fraction is less than 1/b.)